LMFDB - Embedded newform 1275.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,2,Mod(1,1275)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.1809262577$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1275.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} +2.43845 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} +2.43845 q^{8} +1.00000 q^{9} -2.56155 q^{11} +0.438447 q^{12} -4.56155 q^{13} -4.68466 q^{16} -1.00000 q^{17} -1.56155 q^{18} +7.68466 q^{19} +4.00000 q^{22} +6.56155 q^{23} +2.43845 q^{24} +7.12311 q^{26} +1.00000 q^{27} +8.24621 q^{29} -5.12311 q^{31} +2.43845 q^{32} -2.56155 q^{33} +1.56155 q^{34} +0.438447 q^{36} -3.12311 q^{37} -12.0000 q^{38} -4.56155 q^{39} +0.561553 q^{41} +7.68466 q^{43} -1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} -4.68466 q^{48} -7.00000 q^{49} -1.00000 q^{51} -2.00000 q^{52} +4.24621 q^{53} -1.56155 q^{54} +7.68466 q^{57} -12.8769 q^{58} -1.12311 q^{59} +0.876894 q^{61} +8.00000 q^{62} +5.56155 q^{64} +4.00000 q^{66} -4.00000 q^{67} -0.438447 q^{68} +6.56155 q^{69} +10.2462 q^{71} +2.43845 q^{72} -4.24621 q^{73} +4.87689 q^{74} +3.36932 q^{76} +7.12311 q^{78} +15.3693 q^{79} +1.00000 q^{81} -0.876894 q^{82} +9.12311 q^{83} -12.0000 q^{86} +8.24621 q^{87} -6.24621 q^{88} +7.12311 q^{89} +2.87689 q^{92} -5.12311 q^{93} -4.49242 q^{94} +2.43845 q^{96} +11.1231 q^{97} +10.9309 q^{98} -2.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 9 * q^8 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9} - q^{11} + 5 q^{12} - 5 q^{13} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 8 q^{22} + 9 q^{23} + 9 q^{24} + 6 q^{26} + 2 q^{27} - 2 q^{31} + 9 q^{32} - q^{33} - q^{34} + 5 q^{36} + 2 q^{37} - 24 q^{38} - 5 q^{39} - 3 q^{41} + 3 q^{43} + 6 q^{44} - 4 q^{46} + 14 q^{47} + 3 q^{48} - 14 q^{49} - 2 q^{51} - 4 q^{52} - 8 q^{53} + q^{54} + 3 q^{57} - 34 q^{58} + 6 q^{59} + 10 q^{61} + 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} - 5 q^{68} + 9 q^{69} + 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 18 q^{76} + 6 q^{78} + 6 q^{79} + 2 q^{81} - 10 q^{82} + 10 q^{83} - 24 q^{86} + 4 q^{88} + 6 q^{89} + 14 q^{92} - 2 q^{93} + 24 q^{94} + 9 q^{96} + 14 q^{97} - 7 q^{98} - q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 9 * q^8 + 2 * q^9 - q^11 + 5 * q^12 - 5 * q^13 + 3 * q^16 - 2 * q^17 + q^18 + 3 * q^19 + 8 * q^22 + 9 * q^23 + 9 * q^24 + 6 * q^26 + 2 * q^27 - 2 * q^31 + 9 * q^32 - q^33 - q^34 + 5 * q^36 + 2 * q^37 - 24 * q^38 - 5 * q^39 - 3 * q^41 + 3 * q^43 + 6 * q^44 - 4 * q^46 + 14 * q^47 + 3 * q^48 - 14 * q^49 - 2 * q^51 - 4 * q^52 - 8 * q^53 + q^54 + 3 * q^57 - 34 * q^58 + 6 * q^59 + 10 * q^61 + 16 * q^62 + 7 * q^64 + 8 * q^66 - 8 * q^67 - 5 * q^68 + 9 * q^69 + 4 * q^71 + 9 * q^72 + 8 * q^73 + 18 * q^74 - 18 * q^76 + 6 * q^78 + 6 * q^79 + 2 * q^81 - 10 * q^82 + 10 * q^83 - 24 * q^86 + 4 * q^88 + 6 * q^89 + 14 * q^92 - 2 * q^93 + 24 * q^94 + 9 * q^96 + 14 * q^97 - 7 * q^98 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.438447	0.219224
$5$	0	0
$5$	0	0
$6$	−1.56155	−0.637501
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.43845	0.862121
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0.438447	0.126569
$13$	−4.56155	−1.26515	−0.632574	−	0.774500i	$-0.718001\pi$
$13$	−4.56155	−1.26515	−0.632574	+	0.774500i	$0.718001\pi$
$14$	0	0
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.56155	−0.368062
$19$	7.68466	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$19$	7.68466	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	2.43845	0.497746
$25$	0	0
$26$	7.12311	1.39696
$27$	1.00000	0.192450
$28$	0	0
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	2.43845	0.431061
$33$	−2.56155	−0.445909
$34$	1.56155	0.267804
$35$	0	0
$36$	0.438447	0.0730745
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	−12.0000	−1.94666
$39$	−4.56155	−0.730433
$40$	0	0
$41$	0.561553	0.0876998	0.0438499	−	0.999038i	$-0.486038\pi$
$41$	0.561553	0.0876998	0.0438499	+	0.999038i	$0.486038\pi$
$42$	0	0
$43$	7.68466	1.17190	0.585950	−	0.810347i	$-0.300722\pi$
$43$	7.68466	1.17190	0.585950	+	0.810347i	$0.300722\pi$
$44$	−1.12311	−0.169315
$45$	0	0
$46$	−10.2462	−1.51072
$47$	2.87689	0.419638	0.209819	−	0.977740i	$-0.432712\pi$
$47$	2.87689	0.419638	0.209819	+	0.977740i	$0.432712\pi$
$48$	−4.68466	−0.676172
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−2.00000	−0.277350
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	−1.56155	−0.212500
$55$	0	0
$56$	0	0
$57$	7.68466	1.01786
$58$	−12.8769	−1.69082
$59$	−1.12311	−0.146216	−0.0731079	−	0.997324i	$-0.523292\pi$
$59$	−1.12311	−0.146216	−0.0731079	+	0.997324i	$0.523292\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	5.56155	0.695194
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−0.438447	−0.0531695
$69$	6.56155	0.789918
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	2.43845	0.287374
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	4.87689	0.566927
$75$	0	0
$76$	3.36932	0.386487
$77$	0	0
$78$	7.12311	0.806533
$79$	15.3693	1.72918	0.864592	−	0.502475i	$-0.167577\pi$
$79$	15.3693	1.72918	0.864592	+	0.502475i	$0.167577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.876894	−0.0968368
$83$	9.12311	1.00139	0.500695	−	0.865624i	$-0.333078\pi$
$83$	9.12311	1.00139	0.500695	+	0.865624i	$0.333078\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	8.24621	0.884087
$88$	−6.24621	−0.665848
$89$	7.12311	0.755048	0.377524	−	0.926000i	$-0.376776\pi$
$89$	7.12311	0.755048	0.377524	+	0.926000i	$0.376776\pi$
$90$	0	0
$91$	0	0
$92$	2.87689	0.299937
$93$	−5.12311	−0.531241
$94$	−4.49242	−0.463358
$95$	0	0
$96$	2.43845	0.248873
$97$	11.1231	1.12938	0.564690	−	0.825303i	$-0.308996\pi$
$97$	11.1231	1.12938	0.564690	+	0.825303i	$0.308996\pi$
$98$	10.9309	1.10418
$99$	−2.56155	−0.257446
$100$	0	0
$101$	19.1231	1.90282	0.951410	−	0.307927i	$-0.0996352\pi$
$101$	19.1231	1.90282	0.951410	+	0.307927i	$0.0996352\pi$
$102$	1.56155	0.154617
$103$	−4.31534	−0.425203	−0.212602	−	0.977139i	$-0.568194\pi$
$103$	−4.31534	−0.425203	−0.212602	+	0.977139i	$0.568194\pi$
$104$	−11.1231	−1.09071
$105$	0	0
$106$	−6.63068	−0.644029
$107$	−7.68466	−0.742904	−0.371452	−	0.928452i	$-0.621140\pi$
$107$	−7.68466	−0.742904	−0.371452	+	0.928452i	$0.621140\pi$
$108$	0.438447	0.0421896
$109$	−15.1231	−1.44853	−0.724265	−	0.689521i	$-0.757821\pi$
$109$	−15.1231	−1.44853	−0.724265	+	0.689521i	$0.757821\pi$
$110$	0	0
$111$	−3.12311	−0.296432
$112$	0	0
$113$	4.56155	0.429115	0.214557	−	0.976711i	$-0.431169\pi$
$113$	4.56155	0.429115	0.214557	+	0.976711i	$0.431169\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	3.61553	0.335693
$117$	−4.56155	−0.421716
$118$	1.75379	0.161449
$119$	0	0
$120$	0	0
$121$	−4.43845	−0.403495
$122$	−1.36932	−0.123972
$123$	0.561553	0.0506335
$124$	−2.24621	−0.201716
$125$	0	0
$126$	0	0
$127$	0.807764	0.0716775	0.0358387	−	0.999358i	$-0.488590\pi$
$127$	0.807764	0.0716775	0.0358387	+	0.999358i	$0.488590\pi$
$128$	−13.5616	−1.19868
$129$	7.68466	0.676596
$130$	0	0
$131$	18.5616	1.62173	0.810865	−	0.585233i	$-0.198997\pi$
$131$	18.5616	1.62173	0.810865	+	0.585233i	$0.198997\pi$
$132$	−1.12311	−0.0977538
$133$	0	0
$134$	6.24621	0.539590
$135$	0	0
$136$	−2.43845	−0.209095
$137$	16.2462	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$138$	−10.2462	−0.872215
$139$	−9.12311	−0.773812	−0.386906	−	0.922119i	$-0.626456\pi$
$139$	−9.12311	−0.773812	−0.386906	+	0.922119i	$0.626456\pi$
$140$	0	0
$141$	2.87689	0.242278
$142$	−16.0000	−1.34269
$143$	11.6847	0.977120
$144$	−4.68466	−0.390388
$145$	0	0
$146$	6.63068	0.548759
$147$	−7.00000	−0.577350
$148$	−1.36932	−0.112557
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	18.7386	1.51990
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	5.68466	0.453685	0.226843	−	0.973931i	$-0.427160\pi$
$157$	5.68466	0.453685	0.226843	+	0.973931i	$0.427160\pi$
$158$	−24.0000	−1.90934
$159$	4.24621	0.336746
$160$	0	0
$161$	0	0
$162$	−1.56155	−0.122687
$163$	6.87689	0.538640	0.269320	−	0.963051i	$-0.413201\pi$
$163$	6.87689	0.538640	0.269320	+	0.963051i	$0.413201\pi$
$164$	0.246211	0.0192259
$165$	0	0
$166$	−14.2462	−1.10572
$167$	0.807764	0.0625067	0.0312533	−	0.999511i	$-0.490050\pi$
$167$	0.807764	0.0625067	0.0312533	+	0.999511i	$0.490050\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	7.68466	0.587661
$172$	3.36932	0.256908
$173$	18.8078	1.42993	0.714964	−	0.699161i	$-0.246443\pi$
$173$	18.8078	1.42993	0.714964	+	0.699161i	$0.246443\pi$
$174$	−12.8769	−0.976195
$175$	0	0
$176$	12.0000	0.904534
$177$	−1.12311	−0.0844178
$178$	−11.1231	−0.833712
$179$	−9.12311	−0.681893	−0.340946	−	0.940083i	$-0.610747\pi$
$179$	−9.12311	−0.681893	−0.340946	+	0.940083i	$0.610747\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0.876894	0.0648219
$184$	16.0000	1.17954
$185$	0	0
$186$	8.00000	0.586588
$187$	2.56155	0.187319
$188$	1.26137	0.0919946
$189$	0	0
$190$	0	0
$191$	−13.1231	−0.949555	−0.474777	−	0.880106i	$-0.657471\pi$
$191$	−13.1231	−0.949555	−0.474777	+	0.880106i	$0.657471\pi$
$192$	5.56155	0.401371
$193$	24.2462	1.74528	0.872640	−	0.488364i	$-0.162406\pi$
$193$	24.2462	1.74528	0.872640	+	0.488364i	$0.162406\pi$
$194$	−17.3693	−1.24704
$195$	0	0
$196$	−3.06913	−0.219224
$197$	−19.9309	−1.42002	−0.710008	−	0.704194i	$-0.751309\pi$
$197$	−19.9309	−1.42002	−0.710008	+	0.704194i	$0.751309\pi$
$198$	4.00000	0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−29.8617	−2.10106
$203$	0	0
$204$	−0.438447	−0.0306974
$205$	0	0
$206$	6.73863	0.469503
$207$	6.56155	0.456059
$208$	21.3693	1.48170
$209$	−19.6847	−1.36162
$210$	0	0
$211$	−11.3693	−0.782696	−0.391348	−	0.920243i	$-0.627991\pi$
$211$	−11.3693	−0.782696	−0.391348	+	0.920243i	$0.627991\pi$
$212$	1.86174	0.127865
$213$	10.2462	0.702059
$214$	12.0000	0.820303
$215$	0	0
$216$	2.43845	0.165915
$217$	0	0
$218$	23.6155	1.59945
$219$	−4.24621	−0.286932
$220$	0	0
$221$	4.56155	0.306843
$222$	4.87689	0.327316
$223$	−13.9309	−0.932880	−0.466440	−	0.884553i	$-0.654464\pi$
$223$	−13.9309	−0.932880	−0.466440	+	0.884553i	$0.654464\pi$
$224$	0	0
$225$	0	0
$226$	−7.12311	−0.473822
$227$	−23.0540	−1.53015	−0.765073	−	0.643944i	$-0.777297\pi$
$227$	−23.0540	−1.53015	−0.765073	+	0.643944i	$0.777297\pi$
$228$	3.36932	0.223138
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	20.1080	1.32015
$233$	−0.561553	−0.0367885	−0.0183943	−	0.999831i	$-0.505855\pi$
$233$	−0.561553	−0.0367885	−0.0183943	+	0.999831i	$0.505855\pi$
$234$	7.12311	0.465652
$235$	0	0
$236$	−0.492423	−0.0320540
$237$	15.3693	0.998344
$238$	0	0
$239$	10.2462	0.662772	0.331386	−	0.943495i	$-0.392484\pi$
$239$	10.2462	0.662772	0.331386	+	0.943495i	$0.392484\pi$
$240$	0	0
$241$	−21.3693	−1.37652	−0.688259	−	0.725465i	$-0.741625\pi$
$241$	−21.3693	−1.37652	−0.688259	+	0.725465i	$0.741625\pi$
$242$	6.93087	0.445533
$243$	1.00000	0.0641500
$244$	0.384472	0.0246133
$245$	0	0
$246$	−0.876894	−0.0559087
$247$	−35.0540	−2.23043
$248$	−12.4924	−0.793270
$249$	9.12311	0.578153
$250$	0	0
$251$	−24.4924	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$252$	0	0
$253$	−16.8078	−1.05670
$254$	−1.26137	−0.0791452
$255$	0	0
$256$	10.0540	0.628373
$257$	−9.36932	−0.584442	−0.292221	−	0.956351i	$-0.594394\pi$
$257$	−9.36932	−0.584442	−0.292221	+	0.956351i	$0.594394\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	0	0
$261$	8.24621	0.510428
$262$	−28.9848	−1.79069
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	−6.24621	−0.384428
$265$	0	0
$266$	0	0
$267$	7.12311	0.435927
$268$	−1.75379	−0.107130
$269$	20.5616	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$269$	20.5616	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$270$	0	0
$271$	−0.807764	−0.0490682	−0.0245341	−	0.999699i	$-0.507810\pi$
$271$	−0.807764	−0.0490682	−0.0245341	+	0.999699i	$0.507810\pi$
$272$	4.68466	0.284049
$273$	0	0
$274$	−25.3693	−1.53262
$275$	0	0
$276$	2.87689	0.173169
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	14.2462	0.854431
$279$	−5.12311	−0.306712
$280$	0	0
$281$	−19.1231	−1.14079	−0.570394	−	0.821371i	$-0.693210\pi$
$281$	−19.1231	−1.14079	−0.570394	+	0.821371i	$0.693210\pi$
$282$	−4.49242	−0.267520
$283$	3.36932	0.200285	0.100143	−	0.994973i	$-0.468070\pi$
$283$	3.36932	0.200285	0.100143	+	0.994973i	$0.468070\pi$
$284$	4.49242	0.266576
$285$	0	0
$286$	−18.2462	−1.07892
$287$	0	0
$288$	2.43845	0.143687
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.1231	0.652048
$292$	−1.86174	−0.108950
$293$	7.12311	0.416136	0.208068	−	0.978114i	$-0.433282\pi$
$293$	7.12311	0.416136	0.208068	+	0.978114i	$0.433282\pi$
$294$	10.9309	0.637501
$295$	0	0
$296$	−7.61553	−0.442644
$297$	−2.56155	−0.148636
$298$	6.63068	0.384105
$299$	−29.9309	−1.73095
$300$	0	0
$301$	0	0
$302$	−12.4924	−0.718858
$303$	19.1231	1.09859
$304$	−36.0000	−2.06474
$305$	0	0
$306$	1.56155	0.0892680
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	0	0
$309$	−4.31534	−0.245491
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−11.1231	−0.629722
$313$	7.61553	0.430455	0.215228	−	0.976564i	$-0.430951\pi$
$313$	7.61553	0.430455	0.215228	+	0.976564i	$0.430951\pi$
$314$	−8.87689	−0.500952
$315$	0	0
$316$	6.73863	0.379078
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−6.63068	−0.371830
$319$	−21.1231	−1.18267
$320$	0	0
$321$	−7.68466	−0.428916
$322$	0	0
$323$	−7.68466	−0.427586
$324$	0.438447	0.0243582
$325$	0	0
$326$	−10.7386	−0.594758
$327$	−15.1231	−0.836310
$328$	1.36932	0.0756079
$329$	0	0
$330$	0	0
$331$	−6.06913	−0.333590	−0.166795	−	0.985992i	$-0.553342\pi$
$331$	−6.06913	−0.333590	−0.166795	+	0.985992i	$0.553342\pi$
$332$	4.00000	0.219529
$333$	−3.12311	−0.171145
$334$	−1.26137	−0.0690189
$335$	0	0
$336$	0	0
$337$	−32.7386	−1.78339	−0.891694	−	0.452640i	$-0.850482\pi$
$337$	−32.7386	−1.78339	−0.891694	+	0.452640i	$0.850482\pi$
$338$	−12.1922	−0.663170
$339$	4.56155	0.247750
$340$	0	0
$341$	13.1231	0.710656
$342$	−12.0000	−0.648886
$343$	0	0
$344$	18.7386	1.01032
$345$	0	0
$346$	−29.3693	−1.57890
$347$	−24.4924	−1.31482	−0.657411	−	0.753532i	$-0.728348\pi$
$347$	−24.4924	−1.31482	−0.657411	+	0.753532i	$0.728348\pi$
$348$	3.61553	0.193813
$349$	7.43845	0.398171	0.199085	−	0.979982i	$-0.436203\pi$
$349$	7.43845	0.398171	0.199085	+	0.979982i	$0.436203\pi$
$350$	0	0
$351$	−4.56155	−0.243478
$352$	−6.24621	−0.332924
$353$	−22.4924	−1.19715	−0.598575	−	0.801066i	$-0.704266\pi$
$353$	−22.4924	−1.19715	−0.598575	+	0.801066i	$0.704266\pi$
$354$	1.75379	0.0932128
$355$	0	0
$356$	3.12311	0.165524
$357$	0	0
$358$	14.2462	0.752936
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	−9.36932	−0.492440
$363$	−4.43845	−0.232958
$364$	0	0
$365$	0	0
$366$	−1.36932	−0.0715753
$367$	18.2462	0.952444	0.476222	−	0.879325i	$-0.342006\pi$
$367$	18.2462	0.952444	0.476222	+	0.879325i	$0.342006\pi$
$368$	−30.7386	−1.60236
$369$	0.561553	0.0292333
$370$	0	0
$371$	0	0
$372$	−2.24621	−0.116461
$373$	−16.2462	−0.841197	−0.420598	−	0.907247i	$-0.638180\pi$
$373$	−16.2462	−0.841197	−0.420598	+	0.907247i	$0.638180\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	7.01515	0.361779
$377$	−37.6155	−1.93730
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0.807764	0.0413830
$382$	20.4924	1.04848
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	−13.5616	−0.692060
$385$	0	0
$386$	−37.8617	−1.92711
$387$	7.68466	0.390633
$388$	4.87689	0.247587
$389$	−21.8617	−1.10843	−0.554217	−	0.832372i	$-0.686982\pi$
$389$	−21.8617	−1.10843	−0.554217	+	0.832372i	$0.686982\pi$
$390$	0	0
$391$	−6.56155	−0.331832
$392$	−17.0691	−0.862121
$393$	18.5616	0.936306
$394$	31.1231	1.56796
$395$	0	0
$396$	−1.12311	−0.0564382
$397$	−5.36932	−0.269478	−0.134739	−	0.990881i	$-0.543020\pi$
$397$	−5.36932	−0.269478	−0.134739	+	0.990881i	$0.543020\pi$
$398$	−24.9848	−1.25238
$399$	0	0
$400$	0	0
$401$	−6.17708	−0.308469	−0.154234	−	0.988034i	$-0.549291\pi$
$401$	−6.17708	−0.308469	−0.154234	+	0.988034i	$0.549291\pi$
$402$	6.24621	0.311533
$403$	23.3693	1.16411
$404$	8.38447	0.417143
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	−2.43845	−0.120721
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	0	0
$411$	16.2462	0.801367
$412$	−1.89205	−0.0932146
$413$	0	0
$414$	−10.2462	−0.503574
$415$	0	0
$416$	−11.1231	−0.545355
$417$	−9.12311	−0.446760
$418$	30.7386	1.50348
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	28.5616	1.39200	0.696002	−	0.718039i	$-0.254960\pi$
$421$	28.5616	1.39200	0.696002	+	0.718039i	$0.254960\pi$
$422$	17.7538	0.864241
$423$	2.87689	0.139879
$424$	10.3542	0.502843
$425$	0	0
$426$	−16.0000	−0.775203
$427$	0	0
$428$	−3.36932	−0.162862
$429$	11.6847	0.564141
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−4.68466	−0.225391
$433$	−14.3153	−0.687951	−0.343976	−	0.938979i	$-0.611774\pi$
$433$	−14.3153	−0.687951	−0.343976	+	0.938979i	$0.611774\pi$
$434$	0	0
$435$	0	0
$436$	−6.63068	−0.317552
$437$	50.4233	2.41207
$438$	6.63068	0.316826
$439$	−5.75379	−0.274613	−0.137307	−	0.990529i	$-0.543845\pi$
$439$	−5.75379	−0.274613	−0.137307	+	0.990529i	$0.543845\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	−7.12311	−0.338812
$443$	22.8769	1.08691	0.543457	−	0.839437i	$-0.317115\pi$
$443$	22.8769	1.08691	0.543457	+	0.839437i	$0.317115\pi$
$444$	−1.36932	−0.0649849
$445$	0	0
$446$	21.7538	1.03007
$447$	−4.24621	−0.200839
$448$	0	0
$449$	−12.7386	−0.601173	−0.300587	−	0.953755i	$-0.597182\pi$
$449$	−12.7386	−0.601173	−0.300587	+	0.953755i	$0.597182\pi$
$450$	0	0
$451$	−1.43845	−0.0677338
$452$	2.00000	0.0940721
$453$	8.00000	0.375873
$454$	36.0000	1.68956
$455$	0	0
$456$	18.7386	0.877517
$457$	6.80776	0.318454	0.159227	−	0.987242i	$-0.449100\pi$
$457$	6.80776	0.318454	0.159227	+	0.987242i	$0.449100\pi$
$458$	−9.36932	−0.437799
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	−24.9848	−1.16114	−0.580572	−	0.814209i	$-0.697171\pi$
$463$	−24.9848	−1.16114	−0.580572	+	0.814209i	$0.697171\pi$
$464$	−38.6307	−1.79338
$465$	0	0
$466$	0.876894	0.0406213
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	5.68466	0.261935
$472$	−2.73863	−0.126056
$473$	−19.6847	−0.905102
$474$	−24.0000	−1.10236
$475$	0	0
$476$	0	0
$477$	4.24621	0.194421
$478$	−16.0000	−0.731823
$479$	−29.3002	−1.33876	−0.669380	−	0.742920i	$-0.733440\pi$
$479$	−29.3002	−1.33876	−0.669380	+	0.742920i	$0.733440\pi$
$480$	0	0
$481$	14.2462	0.649571
$482$	33.3693	1.51993
$483$	0	0
$484$	−1.94602	−0.0884557
$485$	0	0
$486$	−1.56155	−0.0708335
$487$	7.36932	0.333936	0.166968	−	0.985962i	$-0.446602\pi$
$487$	7.36932	0.333936	0.166968	+	0.985962i	$0.446602\pi$
$488$	2.13826	0.0967945
$489$	6.87689	0.310984
$490$	0	0
$491$	3.36932	0.152055	0.0760276	−	0.997106i	$-0.475776\pi$
$491$	3.36932	0.152055	0.0760276	+	0.997106i	$0.475776\pi$
$492$	0.246211	0.0111001
$493$	−8.24621	−0.371391
$494$	54.7386	2.46281
$495$	0	0
$496$	24.0000	1.07763
$497$	0	0
$498$	−14.2462	−0.638388
$499$	−11.3693	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$499$	−11.3693	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$500$	0	0
$501$	0.807764	0.0360882
$502$	38.2462	1.70701
$503$	25.4384	1.13424	0.567122	−	0.823634i	$-0.308057\pi$
$503$	25.4384	1.13424	0.567122	+	0.823634i	$0.308057\pi$
$504$	0	0
$505$	0	0
$506$	26.2462	1.16679
$507$	7.80776	0.346755
$508$	0.354162	0.0157134
$509$	16.8769	0.748055	0.374028	−	0.927418i	$-0.377977\pi$
$509$	16.8769	0.748055	0.374028	+	0.927418i	$0.377977\pi$
$510$	0	0
$511$	0	0
$512$	11.4233	0.504843
$513$	7.68466	0.339286
$514$	14.6307	0.645332
$515$	0	0
$516$	3.36932	0.148326
$517$	−7.36932	−0.324102
$518$	0	0
$519$	18.8078	0.825569
$520$	0	0
$521$	−31.4384	−1.37734	−0.688672	−	0.725073i	$-0.741806\pi$
$521$	−31.4384	−1.37734	−0.688672	+	0.725073i	$0.741806\pi$
$522$	−12.8769	−0.563606
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	8.13826	0.355522
$525$	0	0
$526$	19.5076	0.850571
$527$	5.12311	0.223166
$528$	12.0000	0.522233
$529$	20.0540	0.871912
$530$	0	0
$531$	−1.12311	−0.0487386
$532$	0	0
$533$	−2.56155	−0.110953
$534$	−11.1231	−0.481344
$535$	0	0
$536$	−9.75379	−0.421300
$537$	−9.12311	−0.393691
$538$	−32.1080	−1.38427
$539$	17.9309	0.772337
$540$	0	0
$541$	−40.1080	−1.72438	−0.862188	−	0.506589i	$-0.830906\pi$
$541$	−40.1080	−1.72438	−0.862188	+	0.506589i	$0.830906\pi$
$542$	1.26137	0.0541803
$543$	6.00000	0.257485
$544$	−2.43845	−0.104548
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	7.12311	0.304284
$549$	0.876894	0.0374249
$550$	0	0
$551$	63.3693	2.69962
$552$	16.0000	0.681005
$553$	0	0
$554$	9.36932	0.398064
$555$	0	0
$556$	−4.00000	−0.169638
$557$	6.49242	0.275093	0.137546	−	0.990495i	$-0.456078\pi$
$557$	6.49242	0.275093	0.137546	+	0.990495i	$0.456078\pi$
$558$	8.00000	0.338667
$559$	−35.0540	−1.48263
$560$	0	0
$561$	2.56155	0.108149
$562$	29.8617	1.25964
$563$	−22.8769	−0.964146	−0.482073	−	0.876131i	$-0.660116\pi$
$563$	−22.8769	−0.964146	−0.482073	+	0.876131i	$0.660116\pi$
$564$	1.26137	0.0531131
$565$	0	0
$566$	−5.26137	−0.221152
$567$	0	0
$568$	24.9848	1.04834
$569$	12.8769	0.539827	0.269914	−	0.962885i	$-0.413005\pi$
$569$	12.8769	0.539827	0.269914	+	0.962885i	$0.413005\pi$
$570$	0	0
$571$	18.7386	0.784187	0.392094	−	0.919925i	$-0.371751\pi$
$571$	18.7386	0.784187	0.392094	+	0.919925i	$0.371751\pi$
$572$	5.12311	0.214208
$573$	−13.1231	−0.548226
$574$	0	0
$575$	0	0
$576$	5.56155	0.231731
$577$	41.0540	1.70910	0.854550	−	0.519370i	$-0.173833\pi$
$577$	41.0540	1.70910	0.854550	+	0.519370i	$0.173833\pi$
$578$	−1.56155	−0.0649520
$579$	24.2462	1.00764
$580$	0	0
$581$	0	0
$582$	−17.3693	−0.719981
$583$	−10.8769	−0.450475
$584$	−10.3542	−0.428458
$585$	0	0
$586$	−11.1231	−0.459491
$587$	−36.9848	−1.52653	−0.763264	−	0.646087i	$-0.776404\pi$
$587$	−36.9848	−1.52653	−0.763264	+	0.646087i	$0.776404\pi$
$588$	−3.06913	−0.126569
$589$	−39.3693	−1.62218
$590$	0	0
$591$	−19.9309	−0.819846
$592$	14.6307	0.601317
$593$	−44.2462	−1.81697	−0.908487	−	0.417913i	$-0.862762\pi$
$593$	−44.2462	−1.81697	−0.908487	+	0.417913i	$0.862762\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−1.86174	−0.0762598
$597$	16.0000	0.654836
$598$	46.7386	1.91128
$599$	−41.6155	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$599$	−41.6155	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$600$	0	0
$601$	34.9848	1.42706	0.713531	−	0.700624i	$-0.247095\pi$
$601$	34.9848	1.42706	0.713531	+	0.700624i	$0.247095\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	3.50758	0.142721
$605$	0	0
$606$	−29.8617	−1.21305
$607$	−15.3693	−0.623821	−0.311911	−	0.950111i	$-0.600969\pi$
$607$	−15.3693	−0.623821	−0.311911	+	0.950111i	$0.600969\pi$
$608$	18.7386	0.759952
$609$	0	0
$610$	0	0
$611$	−13.1231	−0.530904
$612$	−0.438447	−0.0177232
$613$	−2.31534	−0.0935158	−0.0467579	−	0.998906i	$-0.514889\pi$
$613$	−2.31534	−0.0935158	−0.0467579	+	0.998906i	$0.514889\pi$
$614$	−0.768944	−0.0310320
$615$	0	0
$616$	0	0
$617$	27.7538	1.11733	0.558663	−	0.829395i	$-0.311315\pi$
$617$	27.7538	1.11733	0.558663	+	0.829395i	$0.311315\pi$
$618$	6.73863	0.271068
$619$	−19.3693	−0.778519	−0.389259	−	0.921128i	$-0.627269\pi$
$619$	−19.3693	−0.778519	−0.389259	+	0.921128i	$0.627269\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	0	0
$623$	0	0
$624$	21.3693	0.855457
$625$	0	0
$626$	−11.8920	−0.475302
$627$	−19.6847	−0.786130
$628$	2.49242	0.0994585
$629$	3.12311	0.124526
$630$	0	0
$631$	11.6847	0.465159	0.232579	−	0.972577i	$-0.425283\pi$
$631$	11.6847	0.465159	0.232579	+	0.972577i	$0.425283\pi$
$632$	37.4773	1.49077
$633$	−11.3693	−0.451890
$634$	−28.1080	−1.11631
$635$	0	0
$636$	1.86174	0.0738228
$637$	31.9309	1.26515
$638$	32.9848	1.30588
$639$	10.2462	0.405334
$640$	0	0
$641$	−0.0691303	−0.00273048	−0.00136524	−	0.999999i	$-0.500435\pi$
$641$	−0.0691303	−0.00273048	−0.00136524	+	0.999999i	$0.500435\pi$
$642$	12.0000	0.473602
$643$	30.2462	1.19279	0.596397	−	0.802690i	$-0.296598\pi$
$643$	30.2462	1.19279	0.596397	+	0.802690i	$0.296598\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−15.3693	−0.604230	−0.302115	−	0.953271i	$-0.597693\pi$
$647$	−15.3693	−0.604230	−0.302115	+	0.953271i	$0.597693\pi$
$648$	2.43845	0.0957913
$649$	2.87689	0.112928
$650$	0	0
$651$	0	0
$652$	3.01515	0.118083
$653$	4.06913	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$653$	4.06913	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$654$	23.6155	0.923440
$655$	0	0
$656$	−2.63068	−0.102711
$657$	−4.24621	−0.165660
$658$	0	0
$659$	47.8617	1.86443	0.932214	−	0.361907i	$-0.117874\pi$
$659$	47.8617	1.86443	0.932214	+	0.361907i	$0.117874\pi$
$660$	0	0
$661$	25.6847	0.999017	0.499509	−	0.866309i	$-0.333514\pi$
$661$	25.6847	0.999017	0.499509	+	0.866309i	$0.333514\pi$
$662$	9.47727	0.368344
$663$	4.56155	0.177156
$664$	22.2462	0.863320
$665$	0	0
$666$	4.87689	0.188976
$667$	54.1080	2.09507
$668$	0.354162	0.0137029
$669$	−13.9309	−0.538599
$670$	0	0
$671$	−2.24621	−0.0867140
$672$	0	0
$673$	−48.7386	−1.87874	−0.939368	−	0.342910i	$-0.888587\pi$
$673$	−48.7386	−1.87874	−0.939368	+	0.342910i	$0.888587\pi$
$674$	51.1231	1.96919
$675$	0	0
$676$	3.42329	0.131665
$677$	13.6847	0.525944	0.262972	−	0.964803i	$-0.415297\pi$
$677$	13.6847	0.525944	0.262972	+	0.964803i	$0.415297\pi$
$678$	−7.12311	−0.273561
$679$	0	0
$680$	0	0
$681$	−23.0540	−0.883430
$682$	−20.4924	−0.784695
$683$	5.43845	0.208096	0.104048	−	0.994572i	$-0.466820\pi$
$683$	5.43845	0.208096	0.104048	+	0.994572i	$0.466820\pi$
$684$	3.36932	0.128829
$685$	0	0
$686$	0	0
$687$	6.00000	0.228914
$688$	−36.0000	−1.37249
$689$	−19.3693	−0.737912
$690$	0	0
$691$	36.9848	1.40697	0.703485	−	0.710710i	$-0.251626\pi$
$691$	36.9848	1.40697	0.703485	+	0.710710i	$0.251626\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	38.2462	1.45181
$695$	0	0
$696$	20.1080	0.762190
$697$	−0.561553	−0.0212703
$698$	−11.6155	−0.439654
$699$	−0.561553	−0.0212399
$700$	0	0
$701$	−9.36932	−0.353874	−0.176937	−	0.984222i	$-0.556619\pi$
$701$	−9.36932	−0.353874	−0.176937	+	0.984222i	$0.556619\pi$
$702$	7.12311	0.268844
$703$	−24.0000	−0.905177
$704$	−14.2462	−0.536924
$705$	0	0
$706$	35.1231	1.32188
$707$	0	0
$708$	−0.492423	−0.0185064
$709$	4.73863	0.177963	0.0889816	−	0.996033i	$-0.471639\pi$
$709$	4.73863	0.177963	0.0889816	+	0.996033i	$0.471639\pi$
$710$	0	0
$711$	15.3693	0.576394
$712$	17.3693	0.650943
$713$	−33.6155	−1.25891
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	10.2462	0.382652
$718$	3.50758	0.130902
$719$	8.80776	0.328474	0.164237	−	0.986421i	$-0.447484\pi$
$719$	8.80776	0.328474	0.164237	+	0.986421i	$0.447484\pi$
$720$	0	0
$721$	0	0
$722$	−62.5464	−2.32774
$723$	−21.3693	−0.794733
$724$	2.63068	0.0977686
$725$	0	0
$726$	6.93087	0.257229
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.68466	−0.284227
$732$	0.384472	0.0142105
$733$	28.2462	1.04330	0.521649	−	0.853160i	$-0.325317\pi$
$733$	28.2462	1.04330	0.521649	+	0.853160i	$0.325317\pi$
$734$	−28.4924	−1.05167
$735$	0	0
$736$	16.0000	0.589768
$737$	10.2462	0.377424
$738$	−0.876894	−0.0322789
$739$	−8.31534	−0.305885	−0.152942	−	0.988235i	$-0.548875\pi$
$739$	−8.31534	−0.305885	−0.152942	+	0.988235i	$0.548875\pi$
$740$	0	0
$741$	−35.0540	−1.28774
$742$	0	0
$743$	4.49242	0.164811	0.0824055	−	0.996599i	$-0.473740\pi$
$743$	4.49242	0.164811	0.0824055	+	0.996599i	$0.473740\pi$
$744$	−12.4924	−0.457994
$745$	0	0
$746$	25.3693	0.928837
$747$	9.12311	0.333797
$748$	1.12311	0.0410648
$749$	0	0
$750$	0	0
$751$	−0.630683	−0.0230140	−0.0115070	−	0.999934i	$-0.503663\pi$
$751$	−0.630683	−0.0230140	−0.0115070	+	0.999934i	$0.503663\pi$
$752$	−13.4773	−0.491465
$753$	−24.4924	−0.892553
$754$	58.7386	2.13913
$755$	0	0
$756$	0	0
$757$	21.0540	0.765220	0.382610	−	0.923910i	$-0.375025\pi$
$757$	21.0540	0.765220	0.382610	+	0.923910i	$0.375025\pi$
$758$	18.7386	0.680618
$759$	−16.8078	−0.610083
$760$	0	0
$761$	−32.2462	−1.16892	−0.584462	−	0.811421i	$-0.698694\pi$
$761$	−32.2462	−1.16892	−0.584462	+	0.811421i	$0.698694\pi$
$762$	−1.26137	−0.0456945
$763$	0	0
$764$	−5.75379	−0.208165
$765$	0	0
$766$	−16.0000	−0.578103
$767$	5.12311	0.184985
$768$	10.0540	0.362792
$769$	−29.5464	−1.06547	−0.532735	−	0.846282i	$-0.678836\pi$
$769$	−29.5464	−1.06547	−0.532735	+	0.846282i	$0.678836\pi$
$770$	0	0
$771$	−9.36932	−0.337428
$772$	10.6307	0.382607
$773$	33.3693	1.20021	0.600105	−	0.799921i	$-0.295125\pi$
$773$	33.3693	1.20021	0.600105	+	0.799921i	$0.295125\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	27.1231	0.973663
$777$	0	0
$778$	34.1383	1.22392
$779$	4.31534	0.154613
$780$	0	0
$781$	−26.2462	−0.939163
$782$	10.2462	0.366404
$783$	8.24621	0.294696
$784$	32.7926	1.17116
$785$	0	0
$786$	−28.9848	−1.03386
$787$	−6.24621	−0.222653	−0.111327	−	0.993784i	$-0.535510\pi$
$787$	−6.24621	−0.222653	−0.111327	+	0.993784i	$0.535510\pi$
$788$	−8.73863	−0.311301
$789$	−12.4924	−0.444742
$790$	0	0
$791$	0	0
$792$	−6.24621	−0.221949
$793$	−4.00000	−0.142044
$794$	8.38447	0.297554
$795$	0	0
$796$	7.01515	0.248646
$797$	−31.6155	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$797$	−31.6155	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$798$	0	0
$799$	−2.87689	−0.101777
$800$	0	0
$801$	7.12311	0.251683
$802$	9.64584	0.340606
$803$	10.8769	0.383837
$804$	−1.75379	−0.0618514
$805$	0	0
$806$	−36.4924	−1.28539
$807$	20.5616	0.723801
$808$	46.6307	1.64046
$809$	53.0540	1.86528	0.932639	−	0.360810i	$-0.117500\pi$
$809$	53.0540	1.86528	0.932639	+	0.360810i	$0.117500\pi$
$810$	0	0
$811$	−20.6307	−0.724441	−0.362221	−	0.932092i	$-0.617981\pi$
$811$	−20.6307	−0.724441	−0.362221	+	0.932092i	$0.617981\pi$
$812$	0	0
$813$	−0.807764	−0.0283295
$814$	−12.4924	−0.437859
$815$	0	0
$816$	4.68466	0.163996
$817$	59.0540	2.06604
$818$	3.61553	0.126414
$819$	0	0
$820$	0	0
$821$	−16.5616	−0.578002	−0.289001	−	0.957329i	$-0.593323\pi$
$821$	−16.5616	−0.578002	−0.289001	+	0.957329i	$0.593323\pi$
$822$	−25.3693	−0.884857
$823$	−36.4924	−1.27205	−0.636023	−	0.771670i	$-0.719422\pi$
$823$	−36.4924	−1.27205	−0.636023	+	0.771670i	$0.719422\pi$
$824$	−10.5227	−0.366577
$825$	0	0
$826$	0	0
$827$	14.4233	0.501547	0.250774	−	0.968046i	$-0.419315\pi$
$827$	14.4233	0.501547	0.250774	+	0.968046i	$0.419315\pi$
$828$	2.87689	0.0999790
$829$	50.4924	1.75367	0.876837	−	0.480787i	$-0.159649\pi$
$829$	50.4924	1.75367	0.876837	+	0.480787i	$0.159649\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	−25.3693	−0.879523
$833$	7.00000	0.242536
$834$	14.2462	0.493306
$835$	0	0
$836$	−8.63068	−0.298498
$837$	−5.12311	−0.177080
$838$	−50.7386	−1.75274
$839$	11.0540	0.381626	0.190813	−	0.981626i	$-0.438888\pi$
$839$	11.0540	0.381626	0.190813	+	0.981626i	$0.438888\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	−44.6004	−1.53703
$843$	−19.1231	−0.658635
$844$	−4.98485	−0.171585
$845$	0	0
$846$	−4.49242	−0.154453
$847$	0	0
$848$	−19.8920	−0.683096
$849$	3.36932	0.115635
$850$	0	0
$851$	−20.4924	−0.702471
$852$	4.49242	0.153908
$853$	−20.7386	−0.710077	−0.355039	−	0.934852i	$-0.615532\pi$
$853$	−20.7386	−0.710077	−0.355039	+	0.934852i	$0.615532\pi$
$854$	0	0
$855$	0	0
$856$	−18.7386	−0.640473
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	−18.2462	−0.622915
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	37.4773	1.27648
$863$	26.2462	0.893431	0.446716	−	0.894676i	$-0.352594\pi$
$863$	26.2462	0.893431	0.446716	+	0.894676i	$0.352594\pi$
$864$	2.43845	0.0829577
$865$	0	0
$866$	22.3542	0.759625
$867$	1.00000	0.0339618
$868$	0	0
$869$	−39.3693	−1.33551
$870$	0	0
$871$	18.2462	0.618249
$872$	−36.8769	−1.24881
$873$	11.1231	0.376460
$874$	−78.7386	−2.66337
$875$	0	0
$876$	−1.86174	−0.0629023
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	8.98485	0.303224
$879$	7.12311	0.240256
$880$	0	0
$881$	23.7538	0.800285	0.400143	−	0.916453i	$-0.368961\pi$
$881$	23.7538	0.800285	0.400143	+	0.916453i	$0.368961\pi$
$882$	10.9309	0.368062
$883$	−38.4233	−1.29305	−0.646523	−	0.762894i	$-0.723778\pi$
$883$	−38.4233	−1.29305	−0.646523	+	0.762894i	$0.723778\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−35.7235	−1.20015
$887$	−22.5616	−0.757543	−0.378771	−	0.925490i	$-0.623653\pi$
$887$	−22.5616	−0.757543	−0.378771	+	0.925490i	$0.623653\pi$
$888$	−7.61553	−0.255560
$889$	0	0
$890$	0	0
$891$	−2.56155	−0.0858152
$892$	−6.10795	−0.204509
$893$	22.1080	0.739814
$894$	6.63068	0.221763
$895$	0	0
$896$	0	0
$897$	−29.9309	−0.999363
$898$	19.8920	0.663806
$899$	−42.2462	−1.40899
$900$	0	0
$901$	−4.24621	−0.141462
$902$	2.24621	0.0747907
$903$	0	0
$904$	11.1231	0.369949
$905$	0	0
$906$	−12.4924	−0.415033
$907$	−47.8617	−1.58922	−0.794611	−	0.607118i	$-0.792325\pi$
$907$	−47.8617	−1.58922	−0.794611	+	0.607118i	$0.792325\pi$
$908$	−10.1080	−0.335444
$909$	19.1231	0.634273
$910$	0	0
$911$	−29.3002	−0.970758	−0.485379	−	0.874304i	$-0.661318\pi$
$911$	−29.3002	−0.970758	−0.485379	+	0.874304i	$0.661318\pi$
$912$	−36.0000	−1.19208
$913$	−23.3693	−0.773412
$914$	−10.6307	−0.351632
$915$	0	0
$916$	2.63068	0.0869202
$917$	0	0
$918$	1.56155	0.0515389
$919$	−4.31534	−0.142350	−0.0711750	−	0.997464i	$-0.522675\pi$
$919$	−4.31534	−0.142350	−0.0711750	+	0.997464i	$0.522675\pi$
$920$	0	0
$921$	0.492423	0.0162259
$922$	−12.8769	−0.424078
$923$	−46.7386	−1.53842
$924$	0	0
$925$	0	0
$926$	39.0152	1.28212
$927$	−4.31534	−0.141734
$928$	20.1080	0.660076
$929$	31.9309	1.04762	0.523809	−	0.851836i	$-0.324511\pi$
$929$	31.9309	1.04762	0.523809	+	0.851836i	$0.324511\pi$
$930$	0	0
$931$	−53.7926	−1.76298
$932$	−0.246211	−0.00806492
$933$	0	0
$934$	−5.26137	−0.172157
$935$	0	0
$936$	−11.1231	−0.363570
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	7.61553	0.248523
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−8.87689	−0.289225
$943$	3.68466	0.119989
$944$	5.26137	0.171243
$945$	0	0
$946$	30.7386	0.999399
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	6.73863	0.218861
$949$	19.3693	0.628755
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	54.3542	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$953$	54.3542	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$954$	−6.63068	−0.214676
$955$	0	0
$956$	4.49242	0.145295
$957$	−21.1231	−0.682813
$958$	45.7538	1.47824
$959$	0	0
$960$	0	0
$961$	−4.75379	−0.153348
$962$	−22.2462	−0.717247
$963$	−7.68466	−0.247635
$964$	−9.36932	−0.301765
$965$	0	0
$966$	0	0
$967$	−46.5616	−1.49732	−0.748659	−	0.662955i	$-0.769302\pi$
$967$	−46.5616	−1.49732	−0.748659	+	0.662955i	$0.769302\pi$
$968$	−10.8229	−0.347862
$969$	−7.68466	−0.246867
$970$	0	0
$971$	−2.38447	−0.0765213	−0.0382607	−	0.999268i	$-0.512182\pi$
$971$	−2.38447	−0.0765213	−0.0382607	+	0.999268i	$0.512182\pi$
$972$	0.438447	0.0140632
$973$	0	0
$974$	−11.5076	−0.368727
$975$	0	0
$976$	−4.10795	−0.131492
$977$	8.24621	0.263820	0.131910	−	0.991262i	$-0.457889\pi$
$977$	8.24621	0.263820	0.131910	+	0.991262i	$0.457889\pi$
$978$	−10.7386	−0.343384
$979$	−18.2462	−0.583151
$980$	0	0
$981$	−15.1231	−0.482844
$982$	−5.26137	−0.167897
$983$	−2.06913	−0.0659950	−0.0329975	−	0.999455i	$-0.510505\pi$
$983$	−2.06913	−0.0659950	−0.0329975	+	0.999455i	$0.510505\pi$
$984$	1.36932	0.0436522
$985$	0	0
$986$	12.8769	0.410084
$987$	0	0
$988$	−15.3693	−0.488963
$989$	50.4233	1.60337
$990$	0	0
$991$	−6.73863	−0.214060	−0.107030	−	0.994256i	$-0.534134\pi$
$991$	−6.73863	−0.214060	−0.107030	+	0.994256i	$0.534134\pi$
$992$	−12.4924	−0.396635
$993$	−6.06913	−0.192598
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	17.7538	0.561986
$999$	−3.12311	−0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.a.n.1.1		2
3.2	odd	2		3825.2.a.s.1.2		2
5.2	odd	4		1275.2.b.d.1174.2		4
5.3	odd	4		1275.2.b.d.1174.3		4
5.4	even	2		51.2.a.b.1.2	✓	2
15.14	odd	2		153.2.a.e.1.1		2
20.19	odd	2		816.2.a.m.1.1		2
35.34	odd	2		2499.2.a.o.1.2		2
40.19	odd	2		3264.2.a.bg.1.2		2
40.29	even	2		3264.2.a.bl.1.2		2
55.54	odd	2		6171.2.a.p.1.1		2
60.59	even	2		2448.2.a.v.1.2		2
65.64	even	2		8619.2.a.q.1.1		2
85.4	even	4		867.2.d.c.577.2		4
85.9	even	8		867.2.e.f.829.1		8
85.14	odd	16		867.2.h.j.757.4		16
85.19	even	8		867.2.e.f.616.3		8
85.24	odd	16		867.2.h.j.712.2		16
85.29	odd	16		867.2.h.j.688.1		16
85.39	odd	16		867.2.h.j.688.2		16
85.44	odd	16		867.2.h.j.712.1		16
85.49	even	8		867.2.e.f.616.4		8
85.54	odd	16		867.2.h.j.757.3		16
85.59	even	8		867.2.e.f.829.2		8
85.64	even	4		867.2.d.c.577.1		4
85.74	odd	16		867.2.h.j.733.3		16
85.79	odd	16		867.2.h.j.733.4		16
85.84	even	2		867.2.a.f.1.2		2
105.104	even	2		7497.2.a.v.1.1		2
120.29	odd	2		9792.2.a.cy.1.1		2
120.59	even	2		9792.2.a.cz.1.1		2
255.254	odd	2		2601.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.2	✓	2	5.4	even	2
153.2.a.e.1.1		2	15.14	odd	2
816.2.a.m.1.1		2	20.19	odd	2
867.2.a.f.1.2		2	85.84	even	2
867.2.d.c.577.1		4	85.64	even	4
867.2.d.c.577.2		4	85.4	even	4
867.2.e.f.616.3		8	85.19	even	8
867.2.e.f.616.4		8	85.49	even	8
867.2.e.f.829.1		8	85.9	even	8
867.2.e.f.829.2		8	85.59	even	8
867.2.h.j.688.1		16	85.29	odd	16
867.2.h.j.688.2		16	85.39	odd	16
867.2.h.j.712.1		16	85.44	odd	16
867.2.h.j.712.2		16	85.24	odd	16
867.2.h.j.733.3		16	85.74	odd	16
867.2.h.j.733.4		16	85.79	odd	16
867.2.h.j.757.3		16	85.54	odd	16
867.2.h.j.757.4		16	85.14	odd	16
1275.2.a.n.1.1		2	1.1	even	1	trivial
1275.2.b.d.1174.2		4	5.2	odd	4
1275.2.b.d.1174.3		4	5.3	odd	4
2448.2.a.v.1.2		2	60.59	even	2
2499.2.a.o.1.2		2	35.34	odd	2
2601.2.a.t.1.1		2	255.254	odd	2
3264.2.a.bg.1.2		2	40.19	odd	2
3264.2.a.bl.1.2		2	40.29	even	2
3825.2.a.s.1.2		2	3.2	odd	2
6171.2.a.p.1.1		2	55.54	odd	2
7497.2.a.v.1.1		2	105.104	even	2
8619.2.a.q.1.1		2	65.64	even	2
9792.2.a.cy.1.1		2	120.29	odd	2
9792.2.a.cz.1.1		2	120.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} +2.43845 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} +2.43845 q^{8} +1.00000 q^{9} -2.56155 q^{11} +0.438447 q^{12} -4.56155 q^{13} -4.68466 q^{16} -1.00000 q^{17} -1.56155 q^{18} +7.68466 q^{19} +4.00000 q^{22} +6.56155 q^{23} +2.43845 q^{24} +7.12311 q^{26} +1.00000 q^{27} +8.24621 q^{29} -5.12311 q^{31} +2.43845 q^{32} -2.56155 q^{33} +1.56155 q^{34} +0.438447 q^{36} -3.12311 q^{37} -12.0000 q^{38} -4.56155 q^{39} +0.561553 q^{41} +7.68466 q^{43} -1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} -4.68466 q^{48} -7.00000 q^{49} -1.00000 q^{51} -2.00000 q^{52} +4.24621 q^{53} -1.56155 q^{54} +7.68466 q^{57} -12.8769 q^{58} -1.12311 q^{59} +0.876894 q^{61} +8.00000 q^{62} +5.56155 q^{64} +4.00000 q^{66} -4.00000 q^{67} -0.438447 q^{68} +6.56155 q^{69} +10.2462 q^{71} +2.43845 q^{72} -4.24621 q^{73} +4.87689 q^{74} +3.36932 q^{76} +7.12311 q^{78} +15.3693 q^{79} +1.00000 q^{81} -0.876894 q^{82} +9.12311 q^{83} -12.0000 q^{86} +8.24621 q^{87} -6.24621 q^{88} +7.12311 q^{89} +2.87689 q^{92} -5.12311 q^{93} -4.49242 q^{94} +2.43845 q^{96} +11.1231 q^{97} +10.9309 q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 9 * q^8 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9} - q^{11} + 5 q^{12} - 5 q^{13} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 8 q^{22} + 9 q^{23} + 9 q^{24} + 6 q^{26} + 2 q^{27} - 2 q^{31} + 9 q^{32} - q^{33} - q^{34} + 5 q^{36} + 2 q^{37} - 24 q^{38} - 5 q^{39} - 3 q^{41} + 3 q^{43} + 6 q^{44} - 4 q^{46} + 14 q^{47} + 3 q^{48} - 14 q^{49} - 2 q^{51} - 4 q^{52} - 8 q^{53} + q^{54} + 3 q^{57} - 34 q^{58} + 6 q^{59} + 10 q^{61} + 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} - 5 q^{68} + 9 q^{69} + 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 18 q^{76} + 6 q^{78} + 6 q^{79} + 2 q^{81} - 10 q^{82} + 10 q^{83} - 24 q^{86} + 4 q^{88} + 6 q^{89} + 14 q^{92} - 2 q^{93} + 24 q^{94} + 9 q^{96} + 14 q^{97} - 7 q^{98} - q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 9 * q^8 + 2 * q^9 - q^11 + 5 * q^12 - 5 * q^13 + 3 * q^16 - 2 * q^17 + q^18 + 3 * q^19 + 8 * q^22 + 9 * q^23 + 9 * q^24 + 6 * q^26 + 2 * q^27 - 2 * q^31 + 9 * q^32 - q^33 - q^34 + 5 * q^36 + 2 * q^37 - 24 * q^38 - 5 * q^39 - 3 * q^41 + 3 * q^43 + 6 * q^44 - 4 * q^46 + 14 * q^47 + 3 * q^48 - 14 * q^49 - 2 * q^51 - 4 * q^52 - 8 * q^53 + q^54 + 3 * q^57 - 34 * q^58 + 6 * q^59 + 10 * q^61 + 16 * q^62 + 7 * q^64 + 8 * q^66 - 8 * q^67 - 5 * q^68 + 9 * q^69 + 4 * q^71 + 9 * q^72 + 8 * q^73 + 18 * q^74 - 18 * q^76 + 6 * q^78 + 6 * q^79 + 2 * q^81 - 10 * q^82 + 10 * q^83 - 24 * q^86 + 4 * q^88 + 6 * q^89 + 14 * q^92 - 2 * q^93 + 24 * q^94 + 9 * q^96 + 14 * q^97 - 7 * q^98 - q^99

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